Carleson measure estimates for caloric functions and parabolic uniformly rectifiable sets

Abstract

Let $E\subset {\mathbb{R}}^{n+1}$ be a parabolic uniformly rectifiable set. We prove that every bounded solution $u$ to $${\partial }_{t}u-\Delta u=0,\text{in}{\mathbb{R}}^{n+1}\setminus E$$ satisfies a Carleson measure estimate condition. An important technical novelty of our work is that we develop a corona domain approximation scheme for $E$ in terms of regular Lip(1/2,1) graph domains. This approximation scheme has an analogous elliptic version which is an improvement of the known results in that setting.

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Cited by in corpus (1)

• The fractional Lipschitz caloric capacity of Cantor sets